Small shepherd's sundial

Some time ago, I made a simple wooden shepherd's sundial.  It's crude and doesn't work well, but I set it on a lanyard and occasionally use it.  When I decided to use it recently, my wife asked for a small one as a necklace.  Commissioned artwork time!

Since I wanted the sundial to be stylish as well as functional, I thought a bit about materials.  I ended up with a simple design that uses 3/8" copper tubing.  It's the same stuff that you can get at the hardware store for various plumbing jobs, but it's small enough for the task.  I traced out the hour lines using a short python script, and printed them out for use as a guide.

The gnomon is held in a steel cylinder with a slight lip to retain it inside the copper tubing, which has a matching (reverse) lip bored in the end.  You slide the cylinder in from the bottom of the sundial.  Boring the copper was a bit of a problem until I got the boring bit set properly.  I also ended up bursting through the side of the first cylinder because I didn't correctly account for the thickness of the tubing. 

I cut the tubing with a jeweler's saw and filed it to final shape for the noon curve.   The cylinder has two drillings: an axial drilling for the hanging loop and a radial drilling for the gnomon.  The hanging loop is just a piece of brass wire bent into a loop at the top.  I thought of soldering it in place, but decided not to.  I hammered the bottom end of the wire into a small rivet, so the loop is free to turn. 

Here is a trial assembly.

I spent considerable time polishing the copper, the pin, and the cylinder.  The machining was not too arduous, though it did take a few tries, but the engraving was much more tricky.

Here's the final product.

I did not use a watchmaker's square graver (too unwieldy on the round surface) nor the more usual round graver (difficult to get it to bite consistently) to do the engraving.  After a bit of trial on a scrap piece of copper tubing, I found that a very small screwdriver that I had previously sharpened to a narrow cutting blade worked much better for engraving.

To keep everything aligned, I taped the printout to the surface of the copper, and cut through the paper.  This got the copper marked in roughly the right places.  Then I removed the paper and finished the engraving under the microscope.  This took about an hour, and was a bit nerve-wracking because every slip-up is visible.  I'm not too proud of how the engraving came out -- many slips, wiggles, and other awful mishaps are visible if you look closely.  However, it doesn't look embarrassingly bad if you don't use a microscope. 

The sundial certainly looks nice enough on its own, but I am not sure how practical the shiny surface will be in use.  It's cloudy now, which might be for the best!

Astrolabe cases

Two sets of friends are getting married, and as both couples are intellectually curious, I thought an astrolabe would be a nice gift.  But astrolabes alone require some kind of protection... so I decided to make cases for them.

I started from a block of black walnut cut from a tree we had taken down earlier this year.  Since I had previously split it, the wood had been drying all summer in the back yard.

Mostly, the woodworking needed for the boxes was nothing particularly special.  The steps I did were:

1. Rough cut the block of wood into blocks, each about 2 inches larger than my intended final size.  I used a chainsaw for this, and did the work outside.
2. Power planed the blocks so that the edges were square and partially smooth.  Since my power planer makes an enormous mess, I also did this outside.  (The chips from the chainsaw and power planer go in my compost...)
3. Using a large hand crosscut saw, I sliced off the lids of the boxes.  This is a somewhat delicate operation -- even though it requires lots of force -- as you must ensure that the cut is perfectly planar and parallel to (at least one of) the faces of the block.  I don't have a large enough power saw blade, so this is also a manual step!  In all honesty, this is a ripping operation, so I should have used a rip saw... but I don't have one.  My crosscut saw works well enough.
4. I did an initial sanding of the cut surfaces and did a little bit of squaring up of the other faces.
5. I recessed the parts of the box that receive the astrolabe.  This involved tracing the outline of a (disassembled) astrolabe to where the recess needed to go on both the base and the lid of the box.  I then used a router in 1/16" depth increments to cut the recess.  For the base, the recess is stepped since the thumb ring is supported in a more shallow recess than the rest of the astrolabe.  This also gives room for the pointers.  I also recessed the lid.
6. Power belt sanding.  Lots of it!  Starting with 60 grit, I got all faces parallel and square and cleaned off all the cut marks from the saws and planes.  I also rounded the edges where I wanted them round.  Then I repeated the whole process  with 80, 100, 150, and 180 grit.
7. Since I wanted a very fine finish, I then hand sanded the entire box with 220, 320, and 400 grit sandpaper.  Walnut will polish nicely with finer sandpaper if you're planning for an oil finish, but that wasn't necessary for this project.
8. Five coats of polyurethane with plenty of time for drying, and 400 grit hand sanding between each coat.  I made sure that the recesses were left unfinished, since I wanted to adhere a velvet lining.

I wanted to line the inside of the boxes with velvet to protect the astrolabe.  Since the cast acrylic I have been using recently for astrolabes is actually fairly sturdy, a velvet-lined box is surely overkill.  (I did accidentally drop my larger astrolabe, and that shattered the thumb ring.  Since super glue is actually an acrylic, it was a simple matter to fix the crack.  Super glue leaves a mostly invisible joint on acrylic and bonds almost instantly.)

Velvet is a fairly troublesome material for lining.  To make it adhere to a flat surface inside the box, you must support it.  The usual way this is done is to glue the velvet to a sheet of cardboard first, and then glue the cardboard to the box.  Since velvet frays badly, you need to roll the velvet around the back of the cardboard (a second gluing) so that no cut edges show.  Finally, since the boxes have a circular cutout to fit the astrolabe, I had to figure out how to manage that joint.

In the end, I glued the cardboard to the velvet, rolled and glued the edges, but left the circular bottom edge alone.  After the glue dried, I hand stitched the circular bottom edges together.  The result was a stiff velvet "cup" that fits tightly inside the box.

Once the velvet was glued in place, I added hinges and a front clasp.  While hinges and clasps aren't difficult to install, they require precision.  You must be especially careful since you don't want to harm the finish.  I held the box in the vice, but lined it with soft paper towel so that the finish wasn't marred by the vice.  I sharpened (under a microscope) my 1/16" drill bit before starting.  To install the tacks for the clasp, I didn't strike the tacks with the hammer directly, but rather used a recessed punch to ensure that I didn't slip and damage the finish.  When setting the tacks, I also held them in brass tweezers. 

Fluidyne experiments

A fluidyne is a piston-driving heat engine whose only moving part consists of a column of fluid (usually water).  It is a kind of Stirling engine, and it seemed like a neat idea to try to make.  The initial design was invented by C. West, and is described in a nicely written report.

Rather than starting with the "easy" standard design, I tried making a 3-piston version described later in West's report, after describing his initial design.  Here's my rendition.

I did some analysis of this design.  My model sets a displacement for each piston (water column) and temperature for each cylinder (air column).  Half of each cylinder is hot, and half is cold.  The engine is driven into oscillation because the cylinder isn't centered between these two sides, so sometimes it's heating and sometimes it's cooling.  The weight of the pistons acts as restoring force, so assuming they're all of the same length, they have the same resonant frequency.  Surprisingly, the cross sectional area of the cylinders and the length of the cylinders doesn't matter in the analysis.

Here is a typical plot of running frequency as a function of the length of the pistons and temperature difference. 

The engine seems to stall below a certain temperature difference between hot and cold sides, and the stall temperature depends strongly on the coefficient of temperature flow into/out of the cylinders.

The way the engine is supposed to run is that you point a heat gun at the three copper tubes on one side.  These are the hot sides of the cylinders, while the others are cold.

The sad part is that while this engine runs beautifully in theory, I could not get mine to start.  Keeping the temperature of all cylinders the same is critical or the pistons get out of order, and even with that managed, it never started. 

Just to verify that I wasn't completely crazy, I decided to modify the plumbing to make a more standard single piston machine.

I didn't do the analysis of this engine, but it also seemed very unwilling to run unless I flooded the hot cylinders with the pistons. 

Under this condition, the tops of the pistons boiled, which raised the pressure substantially.  Since the water vapor then condensed on the other (cold) side, the piston didn't boil away completely! 

Just to make sure that the fluidyne plumbing wasn't superfluous, I disconnected the plumbing and flooded the hot cylinder on its own. 

Here is a video of this test. 

You can see the top surface of the cold side of the piston oscillating slightly, but much less than when plumbed.  So the fluidyne was "running", even though the boiling seemed very wrong, at least from the original report's description.

However, at this point I recalled reading a rather different design by Morris Dovey.  He described a simple prototype that seemed worth trying, at least by modifying the materials I had at hand.  Although he claims that his design does not depend on resonant frequencies to run, that is dubious at best.  Forced or loaded resonances can vary a bit from their natural frequencies anyway.

I disconnected most of the pipes, added the cold cylinder just hanging off to the side of the hot cylinder, and plugged the cold cylinder with a small screwdriver.

This machine really runs!

It even (sometimes) starts nicely...

But I ran into temperature problems all around, as the clear vinyl tubing is not really able to take the heat from the heat gun.  After a minute or so of running, the tubing invariably springs a leak and the machine stops.

I tried flipping the positions of the hot and cold cylinders -- basically just plugging the end of the hot cylinder and leaving a big air bubble below it -- but this did not work.  I have not done the analysis of Dovey's design to get a more analytical handle on its operating regimes, but it is still definitely a resonant device.  You can feel it try to start running once the heat is sufficient, which is a sure sign that there's a forced resonance.

Clock 4 now runs with intermittent impulsing

"Intermittent" can be a problem, but not in this case!  Based on the numerous power budget calculations I've done, impulsing the pendulum in Clock 4 every minute is too much to ask.  After having gotten the escapement to impulse every period (2 seconds) reliably, with run times around 8 hours, it seemed like the right time to go back to trying to get the intermittent part working again.  Especially, the run times without intermittent escaping were limited by drive cord length -- I had a four fall pulley in place for the clock to run that long. 

Therefore, I added more deep cuts to the count wheel, now five in total.

This means that the escapement should be triggered every 30/5 = 6 pendulum periods, or every 12 seconds.  The pin wheel has 30 pins, so will then have a period of 12 seconds * 30 = 360 seconds = 6 minutes.  The pin wheel is driven through a 1:10 mesh for the drive wheel, so it should make a rotation every hour.  I can therefore drive the minute hand from the drive wheel, although it will run counter clockwise.

With some tuning, the Clock 4 runs with 8 lb of drive weight, directly driving a barrel of 1.2 inches.  The clock's run isn't perfect, as (1) the count wheel double counts immediately following an impulse and (2) sometimes this double-counting skips over an impulse.

But given these issues, Theodore measures the following periods in current configuration:

• 53 seconds for the count wheel

• 4 minutes 24 seconds for pin wheel

Given these measurements the drive barrel will make one rotation about every 44 minutes.  In that time, the weight will have dropped 3.7 inches. 

Thus the power consumed is:

3.7 inches / (12 in/ft)  * 8 lb / (44 min * (60 s/min)) = 9.4 * 10^(-4) ft lb / s = 1.28 mW

More power calculations with Woodward's intermittent grasshopper

By joining the count wheel pusher lever of the the Woodward escapement to the escapement trigger, you can make the escapement trigger once per period.  This is the most frequent that the intermittent grasshopper can be triggered.  Triggering every period already happened by accident, but I decided to force it to occur by linking the mechanisms together without using the count wheel.  This way, I could debug the escapement mechanism... and there were indeed problems there.  I think I've resolved them, and this modified mechanism reliably runs until the weight hits the floor.

Currently, the mechanism runs on 1 lb 14 oz, falling 3.9 inches every 10 minutes.  Converting to standard units, this means that the weight falls

3.9 inches * 25.4 mm/inch / (10 min * 60 s/min) = 0.17 mm/s
1 lb 14 oz = 0.85 kg = 8.3 N

Thus the power consumption is 0.17 mm/s * 8.3 N = 1.37 mW.

This is substantially more pessimistic than my previous figure of 0.325 mW averaged over one minute for the count wheel assembly.  This is even with an improvement resulting from a few changes I made.  The pendulum is now hung from two sharp brass points resting in brass cups.

This new hanger ensures a positive positional lock and a definite axis of rotation for the pendulum with substantially less friction than before.

I also made a number of small improvements including reshaping one of the pin wheel pinion teeth, aligning the impulse hook, and stopping the detent's fall a bit earlier.  Finally, I removed every other pin in the pin wheel, which means that the period of the pin wheel is one minute.

Update: 5/13/2019.
By clipping off the tail of the locking detent to make it somewhat more delicately balanced, I can reduce the drive weight by 6.5 oz.  Thus, the power consumption is

3.9 inches * 25.4 mm/inch / (10 min * 60 s/min) * (1.47 lb * 4.43 N/lb) = 1.08 mW.

Wooden astrolabe

I have wanted an astrolabe for a long time and decided to make one as a small project.  After reading Chaucer's Treatise on the Astrolabe -- which is still a very clear manual for the instrument's use -- I had the plan fixed in my mind.

The finished product works nicely and looks smart.  I can usually measure the time from the stars or sun to within about 10 minutes, and can measure true north within about 5 degrees or so.

As many sources on the internet point out (correctly!) that the astrolabe is a stereographic projection of the sky onto a plane that is tangent to the earth at one of the poles.  For northern hemisphere astrolabes, such as mine, the plane is tangent to the north pole, and the projection point is the south pole.  That makes the north pole (and the north star) the center of the instrument.  Since stereographic projection turns circles on the earth into circles on the projection plane, nearly everything sketched on the astrolabe is also a circle.  For instance, both the equator and the ecliptic, which is the path that the sun appears to move through the sky, are both circles.  Since the ecliptic is almost concentric with the equator, but twisted off the equator by about 23.5 degrees (the tropics!), the ecliptic looks like an offset circle on the astrolabe.

I could do all these projections by geometric constructions, but decided that merely projecting points was easier.  This I did in python, and to keep organized, I chose to design all of the curves and scales in a Jupyter notebook.  The notebook produces SVG files as output that contain the various curves, stars, and scales, all at a fixed scale for printing.  I edited each of the files by hand to add some more difficult annotations or to make aesthetic adjustments.  For instance, the back of the instrument has an equation of time, to which I added some small glosses for "sun fast" and "sun slow" as well as the build date.

The front of the instrument consists of the rete (a simplified star chart), the tympan (a replaceable model of the sky's azimuth and elevation curves for local latitude), and a scale around the outer edge for time and compass directions.  I used this file as the source of my star chart, from which I produced the rete file.

With the rete file in hand, I manually selected the ten brightest stars, and shaped the pointers.  The idea is that the outer two rings go on the body of the instrument, while the rete, proper, starts at the inner two rings.  The picture below is an earlier revision, with somewhat different scales on the rete.  It also contains both front and back pointers.

This earlier revision uses mean solar time, from which the true position of the sun cannot be read directly.  You need to use the equation of time to make this adjustment.  I found that was too error prone.  I prefer to have the front of the astrolabe show the true position of everything, and then correct for mean solar time afterwards if desired. 

I printed two copies of this file, so that I would have clean copies of each for construction.

You need one tympan for each latitude.  This one is for my local latitude.

This file contains the same outer scales as the rete so that the pages can all be scaled the same.  These outer two scales are cut off and disposed, which is why I left some intersections.  The bright red mark is the location of true north, common to all files.

It wasn't too difficult to arrange the lines of constant azimuth and elevation, though I noticed that there is very little documentation about how the "unequal hours" lines are traced.  After playing with the models a bit, I realized that these lines are the horizon line rotated about the local north direction, not rotated about true north. 

The unequal hours aren't particularly in a modern instrument, but were used for reckoning time in Italy until the introduction of weight-driven clocks.  The idea is that day and night are divided into twelve hours of equal length, starting at sunset.  The hours are therefore of unequal length throughout the year.  During the day, the unequal hours can be read from the position of the sun.  At night, the astrolabe is more useful.  By turning the rete so that the stars are oriented correctly, the position of the sun in one of the unequal hours tells you the time.  At least on my instrument, the sketching the unequal hours seemed to occupy unused space in a pleasing way. 

The instrument was built using my usual paper-on-wood scroll saw technique.  I used 1/8" birch plywood for the flat pieces.  The tympan is merely a laminated sheet of paper, so that it is thin and sturdy.  The two pointers were cut from oak. 

Here is the astrolabe disassembled.

The instrument has a brass pin that holds all the parts on the common center (the north pole).  The back pointer has a cutout that sets the pin into place.

This is important because you simultaneously want one edge of the pointer to align with the center of the mounting hole -- so that you can sight across it and then read an elevation on the scale -- and you want the pin there too.  The pin has to fit back into the pointer to give clearance for the sight line.

The front pointer has a similar construction, but I made a small brass button to keep the pin end.  Once the pin is installed, you merely bend the tip of the pin to retain it.  The marks along the front pointer measure declination -- angular distance from the celestial equator.

Finally, I added a thumb ring that sets through a larger pin.  I turned this with a small flourish, and silver soldered the ring closed.

Clock 4 current power consumption

Continuing the thoughts from the previous post... How much power does the current Clock 4 pendulum and count wheel consume?  Especially, how much weight is really necessary to drive it?

I'll treat the pendulum rod and bob as two separate weights...

Rod = 28.86 oz = 0.818 kg, centered at 24" = 0.61 m
Bob = 26.75 oz = 0.758 kg, centered at 45" = 1.14 m

Potential energy for a swinging weight = m g L (1-cos(angle))

The amount of energy at the top of the test swing (4.8 degrees) is

( 0.818 kg * 0.61 m + 0.758 kg * 1.14 m ) * 9.8 N/kg * ( 1 - cos (4.8 degrees) ) = 0.046850 J

At the bottom of the test swing (2.4 degrees), the energy is

( 0.818 kg * 0.61 m + 0.758 kg * 1.14 m ) * 9.8 N/kg * ( 1 - cos (2.4 degrees) ) = 0.011718 J.

Assuming one period of the pendulum is 2 seconds (it's not, but will eventually be):

• The unloaded pendulum takes 65 periods to consume that energy = 0.270 mW
• The pendulum driving the pulling pallet consumes this energy in 54 periods = 0.325 mW
• The complete count wheel assembly consumes this energy in 50 periods = 0.351 mW

We can conclude that

• The count wheel assembly consumes 0.081 mW,
• of which 0.026 mW is due to the backstop.

These power figures are somewhat in line with my previous clocks.  Clock 1 runs on 0.5 mW and Clock 3 runs on 0.8 mW.  So thus far, Clock 3 is more efficient by a bit.

Assume that the escapement is triggered once per minute, is geared through a 10:1 gear mesh, and is driven by a 1" diameter barrel.  How much weight is required for all of these power requirements?

The weight falls at an average speed of pi * 0.0254 m / (36000 s) = 2.216e-6 m/s.

Thus, it takes

• 12.4 kg = 27.4 lb to drive the unloaded pendulum,
• 3.7 kg = 8.2 lb to drive the count wheel (without the pendulum), and
• 16.1 kg = 35.5 lb to drive the pendulum and count wheel assembly.

Way too high, I think!  I need to either improve the pendulum's Q or scrap the idea of the 10:1 gear mesh.

For testing purposes, if I were to drive the clock from the pin escape wheel directly, which has a 3/4" pinion, the weight falls at an average speed of pi * 0.75 in * 0.0254 m/in / (3600 s) = 1.6624e-05 m/s.  The amount of weight necessary to drive the pendulum and count wheel assembly becomes 2.16 kg = 4.8 lb.  (This may not be entirely safe since the pin escape wheel arbor isn't very strong.)

Count wheels and pendulums

Philip Woodward invented many interesting clock mechanisms, which are based around intermittent escaping.  To time the intervals between impulses, his escapements use count wheels.  But he cautions that a poorly-designed count wheel can dramatically alter the Q of the pendulum, and cause reliability problems.  Since my next clock is planned to use Woodward's intermittent grasshopper, impulsing once per minute, I wanted to make sure that the count wheel worked well on its own.

Realizing that my previous clock #1 pendulum has a low unloaded Q because I chose to suspend it in a plain brass pivot, I tried a knife edge suspension.  To hang the pendulum, I took a piece of wood with a branch at a right angle and shaped it into a strong bracket.

The bracket has a slot cut into it to receive the pendulum's knife edge. The pendulum knife edge is crude at this point, and not very artistic.

Unloaded, the Q is around 300-500 with some steel weight I tied onto the bottom. 

For the count wheel, I cut a simple 30 tooth ratchet wheel from 1/8" plywood.  I tried to get the tooth spacing about what would correspond to a degree or two of pendulum amplitude about 6" from the suspension.

The count wheel is driven by two wire lever pallets.

The left pallet attaches to a hole in the pendulum and pulls the count wheel to advance it. The right pallet attaches to a separate anchor point, and serves as the backstop.

Here is the assembly, ready for testing.

Starting from a comfortable amplitude, which pushes the backstop about halfway back, the mechanism will run reliably for somewhat longer than 2 minutes.

Starting from just below an amplitude causing double counting, it will run for over 3 minutes.  This is heartening, because it indicates that impulsing every one minute is feasible, because there is plenty of extra energy available to let off the escapement (not built yet).

Fly for small tent (part 2)

Based on a guy line test fit, guying at the corners is mostly sufficient.

However, the long sides have a tendency to touch the side of the tent.  Adding guys at the middle of each of the long sides seems to resolve this issue.  The only subtlety is the door; the guy line should attach to the left side of the door zipper (when facing the tent) since I'm right-handed.

Parts needed

• 6 stakes
• 6 guy lines each about 2 feet long
• 4 corner attachments
• 2 side attachments for the long sides
• zipper pull

The top of the tent -- where all the seams come together -- is something of a mess.  To fix that, I sewed a rectangular patch at the top.

The six guy point attachments are made from siezed loops of 1/8" braided nylon cord.

These loops are sewn directly into the corners...

... the middle of the back ...

... and the left side of the zipper.

The zipper pull is a bit short, so I added a lanyard.  The lanyard starts as 2 feet of 1/8" nylon cord.

Here is the completed lanyard, which is tied directly onto the zipper pull.

Finally, I added guy lines and set the tent up.  Once up, I sealed the seams!

Fly for a small tent (parrt 1)

We have two small, inexpensive tents that we purchased years ago.  They're simple and easy to set up.  Best of all, they're very lightweight.

But sadly, the rain fly is way too small.  Since it seems that no really suitable tents are on the market, I set about making a replacement fly.

Based on some quick measurements of the tent, the basic idea is that there are four panels.  The front panel is split with a zipper.

The panels lay out on about 6 yards of fabric on a 60" wide roll.

Here's the bill of materials:

• 6 yd of ripstop waterproof nylon.  I'm using 1.1 oz weight, which should lead to about a total of 7 oz.
• 3 foot zipper
• seam sealer
• 4 peg tiedowns

First, I chalked out all of the pieces on the fabric to make sure that everything fit nicely.

The front panel (with the zipper) is handled first so that I can compensate for any issues with the front panel size.  The zipper seam is a little complicated.

I figure that if the overlap (v above) is more than three times the zipper half width (z above), it will be fit nicely.  This means that the seam allowance ought to be about 2 v + 1.5 z.  For my zipper, this was around 5 inches.

Here are the two panels to be zipped together.

Forming the pocket for the zipper.

Sewing the zipper into its pocket.

Completed seam.

The pocket folds over the zipper and is sewn in place.

Duplicated on the back as well.

The zipper doesn't open all the way to the top, so the front and back pockets are pinned together to close the front panels together.

Completed zipper pocket.

Since ripstop nylon is slippery stuff, each of the main seams is sewn in two steps.  (I was trying this seam as an alternative to the usual felled seam.  I did three of the main seams as below, and the last as a felled seam.  The felled seam was much easier, but perhaps wasn't as well aligned.)

First, the fabric is pinned...  (use lots of pins!  the fabric is quite prone to sliding!)

... stay-stitched flat...

... then folded, re-pinned, and final stitched.

Here is the tent with the fly at this point. 

The fly is a bit floppy, but this is not surprising.  Shaping still needs to be done, and guying is critical. The attachment points for the guys are determined experimentally with the tent pitched.

Still to come:

• Guy lines
• Stitch the bottom edge
• Shape the top
• Clean up loose thread ends
• Seal the seams
• Add a nice zipper pull